6.2 Clustering
6.2.6 Influence of Reynolds number
Up to this point, all simulations were done for a constant Taylor-based Reynolds number Rλ ≈ 48. In this subsection we intend to investigate the influence of the Reynolds number onto the nematic-isotropic transition and onto the formation of patches. All simulations presented here are for point-like particles in a Kraichnan flow field with number densityρ3 = 1 and varying numbers of particles from 303 to 1003. Hence, the minimum wavenumber (being bound by the box size) varies while the maximum wavenumber iskmax= 30 for all simulations. The smallest length scale in the turbulent field is thus the same for all simulations. However, the resulting integral length scale of the vorticity is not the same for all simulations but necessarily increases slightly in the range 1.03≤ Lω11/ ≤ 1.19 because the integral length scale is a function of the ratio of the maximum to the minimum wavenumber (see Eq.
3.21b). This interval of integral length scales is well within the range where we expect an enhanced patch concentration due to the turbulent field as discussed in the previous Section 6.2.5. We do not expect this small change to have an effect onto the results. The nematic-isotropic transition and the clustering of these simulations with varying Reynolds number are shown in Fig. 6.13. The curves of the nematic order parameter collapse for small vortical Stokes numbers with speed correction, i.e. in the isotropic phase. The isotropic-nematic transition occurs around the same vortical Stokes number and all simulations reach a similarly high global nematic order parameter in the nematic phase. However, the data suggest that the value ofS deep in the nematic phase decreases with increasing Reynolds number. The curves of|Q|, on the other hand, all collapse within their accuracy for high vortical Stokes numbers and all show the “sweet spot” where small patches are observed. But the degree of clustering measured by|Q|in the isotropic phase decreases with increasing turbulent Reynolds number. To conclude, the main finding, which is the “sweet spot” in |Q|, is preserved in the investigated range of turbulent Reynolds numbers.
6.3 Discussion
The nonequilibrium phase diagrams of all three models show two distinct phases: An isotropic phase and a nematic phase. The isotropic phase is present in two domains of the phase diagram (P < 1 and for small Sω,v) because there exist two different and independent mechanisms to destroy nematic order in the model. The system cannot develop nematic order if the stochastic noise is dominant in comparison to the strength of the nematic alignment. Even if two particles meet and tend to align in one time step, their orientations will be effectively randomized by the stochastic noise in the next time step. The second mechanism which prevents the system from globally aligning is the turbulent field. If the vorticity of the turbulent field is larger than the
Rλ= 178.3 Rλ= 152.6 Rλ= 124.8 Rλ= 109.5 Rλ= 93.6 Rλ= 76.0
nematicorderparameterS
vortical Stokes number with speed correction Sω,v
10−1 100 101 102
0 0.2 0.4 0.6 0.8 1
(a) Isotropic-nematic transition.
patchenhancementfactor|Q|
vortical Stokes number Sω
10−2 10−1 100 101
10−2 100 102 104
(b) Patch concentration enhancement factor.
Figure 6.13: Influence of Taylor-based Reynolds number on simulations with point particles in the Kraichnan flow field (to be published in Breier et al., 2017, supple-mentary information).
6.3 Discussion
Parameter Kraichnan flow DNS flow
Rλ 48 17
kmax/kmin= 16 L11/ηK = 10.8
ωK 4.54 4.53
urms 0.94 1.29
2π/kmax/= 1.86 ηK/= 0.30
Table 6.2: Comparison between flow fields from DNSs and kinematic simulations (using the standard parameters).
nematic relaxation constant, the turbulent field acts in the same way as a noise and effectively randomizes the orientations of the particles. The difference between the turbulent field and the true stochastic noise is that the turbulent field is correlated in time and space. The critical vortical Stokes number with speed correction for the isotropic-nematic transition is Sω,vcrit = 1 for the particles (point-like or extended) in the Kraichnan flow field. However, it is ratherSω,vcrit = 0.2 for the point particles in the DNS flow field. The main parameters, which characterize the flow fields, are given in Tab. 6.2. They match roughly in Kolmogorov shear rate and root-mean-square velocity. The Taylor-based Reynolds numbers of the two approaches are comparable;
the kinematic simulations are performed at a largerRλ, though. A direct comparison between the two has to be taken with a grain of salt, because the definition of Rλ
in a Kraichnan flow field is difficult (see discussion in Section 3.1.4). However, the ratio between the largest and smallest length scales in the two simulation approaches is of the same order of magnitude which makes them comparable. Nevertheless, the ratio between smallest length scale of the turbulent field and nematic interaction range differs substantially. For the Kraichnan flow field the smallest wavelength is almost twice as large as the nematic interaction range. In contrary, the Kolmogorov length scaleηK of theDNSflow field is only a third of the nematic interaction range.
The Kolmogorov length scale serves to estimate the size of the smallest structures in the flow field. However, this difference cannot explain why the isotropic-nematic transition occurs at a value of Sω,v smaller than unity when a DNS flow field is used, because a smaller Kolmogorov length scale should rather destroy than stabilize nematic order.
All three models show a comparable behavior of the clustering as measured by|Q|.
The clustering of particles is clearly enhanced in the nematic phase of our system as compared to the isotropic phase. Moreover, the clustering shows a non-monotonic be-havior when the Stokes number is increased. Hence, a specific ratio between turbulent vorticity and nematic interaction strength favors the formation of small patches. To understand this behavior, we discuss theformation of clusters. We know that we need nematic interaction for the cluster formation since clusters are only forming in the
ne-matic phase. Moreover, the self-propulsion is important because non-motile particles would just act as passive tracers. The turbulent flow field is incompressible and hence volume-preserving. This means that the particles will always remain homogeneously distributed, if they followed a homogeneous distribution at one point in time. All presented simulations start from a homogeneous distribution of particles and hence the self-propulsion is necessary to break the volume conservation and form clusters.
The temporal evolution of the clustering system shows that the system firstly orders nematically and only then the clusters are forming. The mechanism is the following:
In the beginning of a simulation all particles are randomly distributed with random orientations. They move forward due to their self-propulsion and are advected by the turbulent field which simply acts as a perturbation of their self-propulsion veloc-ity in this initial phase. Moreover, their orientations are subject to stochastic noise and turbulent vorticity where also the latter acts in this early stage as an additional noise. If the time scale of the nematic interaction is smaller than the time scale of rotational diffusion (1/γ <1/Dr) , the particles can order nematically rather quickly.
This condition is equal to P > 1. Moreover, also the turbulent field must not be stronger than the nematic interaction which is equal to the condition Sω,v >1. Once the particles are ordered nematically, their orientations are altered by the turbulent vorticity and the stochastic noise. We assume the turbulent vorticity to be larger than the stochastic noise (Sω <P) to be able to neglect the stochastic noise and focus on the influence of the turbulent field. This assumption is necessary to study the effect of the turbulent field and corresponds to the area in the |Q|(Sω,P)-plot above the black line (see Fig. 6.4a). So the nematically ordered particles are moving into the direction of their intrinsic orientations and are advected by the turbulent field.
Moreover, the turbulent vorticity alters their orientations and makes them different from the orientations of the neighboring particles. This alteration thus increases the probability of a particle to bump into another particle. If particles meet, they align.
They can form (the beginning of) a cluster if they align polarly and they are subject to roughly the same turbulent field so that the turbulent field does not tear them apart. This last condition is quantified by the integral length scale of the turbulent field. We showed that the integral length scale of the vorticity has to be larger than a quarter of the nematic interaction for a “sweet spot” in the clustering to occur.
If the integral length scale is smaller, the structures of the turbulent field are large enough so that the particles will be effectively torn apart by the turbulent vorticity.
The clustering is always enhanced in the nematic phase as compared to the isotropic phase. This enhancement is due to the fact that the nematic order helps particles to stay close to each other. However, if the integral length scale of the vorticity is larger than a quarter of the nematic interaction range, the strongest clustering should occur if therelevant time scaleof the turbulent field and the time scale of the nematic interaction match. The relevant time scale of the turbulent field is the Kolmogorov
6.3 Discussion
hν
Na K
MgCa Si Fe
not to scale
Figure 6.14: Illustration of the consequence of turbulence induced clustering on differ-ent phytoplankton species (cf. Fig.1.4). The images of the species have been adapted fromHaeckel(1899, online: http://biolib.mpipz.mpg.de/haeckel/kunstformen/
natur.html, copyright 1999 Kurt Stueber und Max-Planck-Institut für Züchtungs-forschung).
time scaleτη =qν/which is related via= 2νR k2E(k) dkto the Kolmogorov shear rate as τη = 1/ωK. A vortical Stokes numberSωc = 2 hence means that the two times equal and this is where the maximum in|Q|was found. To conclude, this means that turbulence can increase the small patches of the system if the integral length scale of the vorticity is larger than a quarter of the nematic interaction range. The maximum increase is found where the Kolmogorov time scale matches the time scale of the nematic alignment. This precisely matches the results of Wang and Maxey (1993) who found that“the strongest accumulation happens when the particle response time is comparable to the Kolmogorov timescale” in a system of settling spherical particles in homogeneous turbulence. More recently, a maximum in clustering of gyrotactic particle in turbulence was also found by Durham et al.(2013) when the Kolmogorov time scale and the typical reorientation time of the particles are equal. This balance of time scales seems to be universal to all systems of particles which are subject to a turbulent field.
The hydrodynamic interaction of planktonic microorganisms is to first order ne-matic (see Section2.1andBaskaran and Marchetti,2009). If they exist in a turbulent environment, the maximum clustering will occur where twice the integral length scale
of the turbulent field matches thenematic interaction range. In principle the nematic interaction range is just a model parameter in our system of self-propelled particles.
Our model is justified by the symmetry of hydrodynamically interacting microswim-mers (see Section2.1) and so the nematic interaction range can be related to physical quantities of the more realistic microswimmers behind. For microswimmers which are modeled as a rigid dumbbell the active volume is given by Vactive = l¯a(vl2/D) with l being the length of the dumbbell, ¯a = (aL +aS)/2 being the mean size of the two spheres, andD being the diffusion constant (Baskaran and Marchetti,2009).
The idea behind the dumbbell model is a stroke-averaged microswimmer so that the two spheres represent the cell body and the volume where the flagella perform their stroke motion. It is important to stress here that the velocity v is not the self-propulsion speed of the particle but is the convective velocity which is proportional to the strength of the force dipole divided by the friction of the dumbbell in the ambient fluid. We can conclude for our model that the nematic interaction range depends not only on the length of the swimmer, but also on the size of the cell body, the stroke radius, and the strength of the corresponding force dipole. The diffusion also plays a role as well as friction due to the ambient fluid. This interpretation of the nematic interaction range means that different organisms will be subject to clus-tering in different turbulent flows, i.e. turbulent flows with different integral length scales of the vorticity. This is a possible route to solve the “paradox of the plankton”
(Hutchinson, 1961) because it favors the clustering of different kinds of organisms in different regions. This spatial separation of different species hence increases the possibility of e.g. sexual mating for a given species because ideally only individuals of the same species cluster in one region in space. Different species can thus coex-ist in the same water body because they are effectively separated in space. Coming back to the illustration of the situation of phytoplankton in the ocean (see Fig. 1.4 in the introduction of this thesis), the situation is the following (see Fig.6.14 for an illustration): A turbulent flow field can be induced in the water e.g. by wind (blue arrow). If the integral length scale of the vorticity does differ in different parts of the water (depicted by differently sized spirals), the different species can cluster in different regions according to their size. This separates the different species in space so that they can coexist.
Even for species with a similar active volume the clustering itself is possibly enough to lead to species diversity. Such a plankton population with different species of similar size leads to species-diverse clusters in the first step. If the clusters are stable compared to the typical time scale in the reproduction cycle, in each cluster a single species might survive due to competitive exclusion. However, because the small-scale conditions vary between clusters different species might survive in different clusters so that we globally find coexistence of competing species. This interpretation does not even need the formation of different turbulent field with different integral length scales of the vorticity.
7 Conclusion and Outlook
7.1 Conclusion
This thesis posed the following research questions: What kind of patterns can emerge in three-dimensional suspensions of self-propelled, aligning, low-Reynolds number swimmers mimicking entities like bacteria and how do these patterns fit in the steady-state phase space? Are simple physical interactions enough to trigger the formation of complex patterns like propagating waves? How can the symmetry of the system be broken? What is the influence of an external flow field on the properties of large groups of self-propelled particles? Or asked differently, how does turbulence influence large groups of motile plankton in their natural habitats – lakes and oceans?
To address these questions we perform large-scale molecular dynamics simulations to solve the corresponding equations of motion. The individual particles align nemat-ically and are subject to rotational diffusion. They are self-propelled with a constant speed. An external flow field is modeled via the method of kinematic simulations to acquire reasonable computation times and a turbulence-like behavior. This external flow field advects and rotates the particles according to the local velocity and vortic-ity. We also include the possibility of a finite size of the particles resulting in steric interactions among individuals.
The system under investigation is intrinsically out of equilibrium because each self-propelled particle moves at a constant speed while it still interacts with its neighbors.
This is only possible if energy is constantly converted into motion from some inter-nal reservoir. Even though our model does not include this process explicitly, it is the reason why we refer to our system as a nonequilibrium system. Almost all of
our simulations start from the most disordered state one might think of: The par-ticles’ positions are randomized in the simulation domain and the orientations are distributed randomly as well. This is what we call a homogeneous and isotropic state. In the temporal evolution the system undergoes a transient evolution until it reaches a steady state where the configuration of both positions and orientations is temporally stable.
We observe different kinds of symmetry breaking depending on the system param-eters like number density ρ and rotational Péclet number P. The latter serves to compare the strength of nematic alignment and with the rotational noise. With in-creasing density and Péclet number therotational symmetry of the systems is broken and global nematic alignment occurs. At low densities, this transition is accompanied by a coexistence state of nematically ordered and isotropic domains in the system.
Such a coexistence leads to an inhomogeneous density distribution because particles accumulate in the nematic domains and as a result the spatial symmetry is broken.
Deep in the nematic phase a spontaneous chiral symmetry breaking occurs through the formation of helices of the local nematic directors. It can be shown that this pattern is indeed a stable configuration of the underlying equations of motion and forms due to the delicate interplay between rotational fluctuations and alignment.
In a one-dimensional system of non-moving rotors under a rapid quench (similar to the XY-model) such a chiral symmetry breaking can also occur in terms of trapped spin waves. When the chiral symmetry is broken the system of SPPs also shows oscillations in the global polarization and hence a breaking of thenematic symmetry.
This is coupled to a density wave traveling along the helical axis so that the particles are inhomogeneously distributed. A symmetry breaking ofnematic symmetry is also observed at small densities close to the isotropic-nematic transition where the sys-tem exhibits propagating waves which are locally polarly ordered. At the same time the homogeneity of the system is broken because the waves appear in a soliton-like fashion with ordered, highly dense domains interrupted by a isotropic, dilute gas.
Such waves were observed in two dimensional systems before and could be shown to be the outcome of an instability. We find that similar waves also occur in three dimensions and with nematic (instead of polar) alignment. In systems with strong nematic interaction and at low global densities the system of self-propelled particles even exhibits a breaking of the global nematic symmetry with the onset of global polar order. The local interaction between particles leads to the coupling of strong alignment and large local density because polarly aligned particles stick together if the Péclet number is large enough and the alignment mechanism is fast compared to the flight time of a particle through the interaction range. A nematic domain is formed by two such counter-propagating subgroups of self-propelled particles.
To address the question of the influence of a turbulent flow field onto motile plank-ton, we model self-propelled, aligning particles (as before) in an external turbulent flow field. Global alignment occurs if the stochastic noise and the turbulent field
7.1 Conclusion
are weak compared to the nematic interaction strength. Our main finding is that turbulence can enhance clustering in terms of the formation of small-scale patches of (self-propelled) particles. The conditions for cluster formation are: (i) the system is in the nematic state, and (ii) the integral length scale of the vorticity is comparable to (at least a quarter of) the nematic interaction range. The strongest clustering occurs if the relevant length and time scales match: twice the integral length scale of the
are weak compared to the nematic interaction strength. Our main finding is that turbulence can enhance clustering in terms of the formation of small-scale patches of (self-propelled) particles. The conditions for cluster formation are: (i) the system is in the nematic state, and (ii) the integral length scale of the vorticity is comparable to (at least a quarter of) the nematic interaction range. The strongest clustering occurs if the relevant length and time scales match: twice the integral length scale of the